NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image....
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Transcript of NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image....
![Page 1: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/1.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Lecture 14
• Reprise: dirty beam, dirty image.
• Sensitivity
• Wide-band imaging
• Weighting– Uniform vs Natural– Tapering– De Villiers weighting– Briggs-like schemes
![Page 2: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/2.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Reprise: dirty beam, dirty image.• Fourier inversion of V times the sampling
function S gives the dirty image ID:
• This is related to the ‘true’ sky image I´ by:
• The dirty beam B is the FT of the sampling function:
• (Can get B by setting all the V to 1, then FT.)
vmulivuSvuVdvdumlI 2D e,,,
mlBmlImlI ,,,D
vmulivuSdvdumlB 2e,,
![Page 3: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/3.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Reprise: l and m• Remember that l = sin θ. θ is the angle from
the phase centre.
• For small l, l ~ θ (in radians of course).• m is similar but for the orthogonal direction.
Direction of phase centre.
Direction ofsource.
l
θ
![Page 4: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/4.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Sensitivity• Image noise standard deviation (for the weak-
source case) is (for natural weighting)
• N here is the number of antennas.
• Note that Ae is further decreased by correlator effects – for example by 2/π if 1-bit digitization is used.
• Actual sensitivity (minimum detectable source flux) is different for different sizes of source.– Due to the absence of baselines < the minimum
antenna separation, an interferometer is generally poor at imaging large-scale structure.
tNN
T
A
kII
1
2 total
erms
![Page 5: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/5.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
How can we increase UV coverage?…we could get more baselines if we moved the antennas!
Wide-band imaging.
![Page 6: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/6.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
…but it is simpler to change the observing wavelength.
eg
λ
λ/2
![Page 7: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/7.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
…we have many baselines,
and, effectively,
many antennas.
With many wavelengths…
![Page 8: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/8.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
16 x 1 MHz 2000 x 1 MHz
Merlin, δ=+35° eMerlin, δ=+35°
Narrow vs broad-band: UV coverage
![Page 9: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/9.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
16 x 1 MHz 2000 x 1 MHz
Narrow vs broad-band - without noise:
![Page 10: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/10.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
SNR of each visibility = 15%.
16 x 1 MHz 2000 x 1 MHz
Narrow vs broad-band - with noise:
![Page 11: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/11.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Weighting: or how to shape the dirty beam.
• Why should we weight the visibilities before transforming to the sky plane?– Because the uneven distribution of samples of V means that the dirty beam has lots of ripples or sidelobes, which can extend a long way out.
• These can hide fainter sources.
– Even if we can subtract the brighter sources, there are always errors in our knowledge of the dirty beam shape.
• If there must be some residual, the smoother and lower it is, the better.
![Page 12: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/12.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Weighting
• There are usually far more short than long baselines.
Baseline length
The distribution of baselinesalso nearly always hasa ‘hole’ in the middle.
![Page 13: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/13.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Weighting• A crude example:
This bin has 1 sample.
This bin has 84 samples.
![Page 14: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/14.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Weighting• What do we get if we leave the visibilities alone?
– The resulting dirty beam will be broad ( low resolution), because there are so many more visibility samples at small (u,v) than large (u,v).
– BUT, if the uncertainties are the same for every visibility, leaving them unweighted (ie, all weights Wj,k=1) gives the lowest noise in the image.
– This is called natural weighting.
• The easiest other thing to do is set Wj,k=1/(the number of visibilities in the j,kth grid cell).– This is called uniform weighting.
• Then optionally multiply everything by a Gaussian:– Called tapering.
![Page 15: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/15.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Natural weighting Uniform weighting
Natural vs uniform:
![Page 16: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/16.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Natural weighting Uniform weighting
The resulting dirty images:
![Page 17: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/17.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
SNR of each visibility = 0.7%.
Natural weighting Uniform weighting
But if we add in some noise...
![Page 18: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/18.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Tradeoff
• This sort of tradeoff, between increasing resolution on the one hand and sensitivity on the other, is unfortunately typical in interferometry.
![Page 19: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/19.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Some other recent ideas:1. Scheme by Mattieu
de Villiers (new, not yet published SA work):– Weight by inverse of
‘density’ of samples.
2. My own contribution:– Iterative optimization.
Has the effect of rounding the weight distribution to ‘feather out’ sharp edges in the field of weights.
– Haven’t got the bugs out of it yet.
Ideal smooth weight function(Fourier inverse of desired PSF)
Isolated samplesget weighted higherso that the averageapproaches the ideal.
Densely packedsamples aredown-weighted.
![Page 20: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/20.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Uniform
Tapered uniform
Iterative best fit out-side 20-pixel radius
Simulated e-Merlin data.400 x 5 MHz channels;νav = 6 GHz;tint = 10 s;δ = +30°
Weighting schemes:
![Page 21: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/21.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
‘Dirty beam’ images (absolute values).
20
Iterative best fit out-side 20-pixel radius
Tapered uniform
Uniform
![Page 22: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/22.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
Natural
Uniform
Optimized
Natural (narrow-band)
Natural
Uniform
Optimized for r>10
Comparison – slices through the DIs:
![Page 23: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/23.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
r = 10
More on iterated weights:
![Page 24: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/24.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
SNR of each visibility = 5.
But real data is noisy…
![Page 25: NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 14 Reprise: dirty beam, dirty image. Sensitivity Wide-band imaging Weighting –Uniform vs Natural.](https://reader036.fdocuments.us/reader036/viewer/2022070411/56649f4f5503460f94c71dd2/html5/thumbnails/25.jpg)
NASSP Masters 5003F - Computational Astronomy - 2009
1. Multiply visibilitieswith a vignettingfunction of time andfrequency, eg
2. Aips task IMAGRparameter UVBOX:effectively smoothsthe weight function.See also D Briggs’PhD thesis.
One could think of other ‘feathering’ schemes.